In this paper, we study the effects of the spatial covariant gravities on the propagation of GWs, produced by the coalescence of compact binaries, and their observational constraints with observational data of GW events from LIGO/Virgo/KAGRA collaboration. For this purpose, we calculate the effects of the spatial covariant gravities to the GW friction, speed of the GWs, amplitude and velocity birefringences, and modified dispersion relation during the propagation of GWs in the cosmological background. These effects can be unified described by a universal parametrization proposed in [1]}, [2]}. Different effects correspond to different parameters, as discussed in details in Sec. III and summarized in Table. REF . These effects can be divided into three classes: (1) the frequency-independent effects which include the modification to the speed of GWs and GW friction; (2) the parity-violating effects which include the amplitude and velocity birefringences of GWs; and (3) the Lorentz-violating effects which include the modified damping rate and dispersion relation of GWs. Among these effects, the parity-violating and Lorentz-violating effects are frequency-dependent. Depending on different coefficients in the spatial covariant gravities, these frequency-dependent effects can produce amplitude modulation and phase corrections to the waveform of GWs produced by the coalescence of compact binaries. The calculation of these modified GW waveform are presented in Sec. III. These modified waveforms provide important tools for constraining parity violating or Lorentz violating effects in the spatial covariant gravities with current available GW events or future detections of GWs.
